On a Strong Precompactness of Velocity Averages for a Heterogenous Transport Equation with Rough Coefficients

We consider strong Lloc(IR d) precompactness of the sequence of averaged quantities ∫ IRm hn(x, λ)ρ(λ)dλ, where ρ ∈ C0(IR), and hn ∈ L p loc(IR d× IRm), p > 1, are solutions to the transport equations with flux explicitly depending on space: divx(F (x, λ)hn(x, λ)) = d ∑ i=1 ∂xi∂ ki λ G i n(x, λ), x ∈ IR, λ ∈ IR, where F = (F1, ..., Fd), and Fi ∈ Lqloc(IR × IRm), i = 1, ..., d, 1/p + 1/q < 1, and ki = (k i 1, ..., k i m) ∈ INm, i = 1, . . . d, stands for multindex. For the sequences of functions (Gn)n∈IN , i = 1, . . . , d, we assume that they strongly converge to zero as n →∞ in Lq̃loc(IR × IRm) for a q̃ > 1. In order to obtain the result we adapt H-measures [13, 35] and give positive (but partial) answer on the question whether it is possible to translate in an algebraic way the information ”uk is bounded in L p, Puk is bounded in L q for a p, q > 1”, where P is a differential operator. This question was posed in [13]. The proof mostly involves the theory of multipliers.